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If the Saturn V rocket were attached to the ground facing in the opposite direction of the Earth's spin, would it in anyway slow the Earth's spin on its axis? How significant of an impact would it have?

Is it possible to measure how much it would slow down the Earth's rotation—assuming it does, of course?

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The Saturn V threw roughly $m=3\times 10^6{\rm kg}$ out its hinder end at a speed of about $v=3{\rm km\,s^{-2}}$. The angular momentum of this mass thrown tangentially to the ground about Earth's center is then $R_\oplus\,m\,v$, where $R_\oplus$ is the Earth's radius. Assuming the Earth to be uniformly dense for a rough figure, its mass moment of inertia about its center is $I=\frac{2}{5}\,M_\oplus\,R_\oplus^2$. If $\Delta\omega$ is the angular speed change wrought by the firing of the rocket, then conservation of system angular momentum is approximately written:

$$I\,\Delta\omega = \frac{2}{5}\,M_\oplus\,R_\oplus^2\,\Delta\omega = R_\oplus\,m\,v$$.

Solve this equation for $\Delta\omega$: it is not large! It will show you that the effect is utterly negligible.

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If you use a highly simplified model without an atmosphere, it could slow the rotation of the Earth. It would be like riding a ferris-wheel and throwing a piece of popcorn straight ahead as you go around. AKA negligible.

Physically speaking the torque created by the rocket, assuming its bolted down well enough, would be the force it exerts times the radius of the Earth. Then multiply that by the time the rocket fires for to get the change in angular momentum of the Earth.

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